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MODELING PROPAGATION PROCESSES ON
NETWORKS BY USING DIFFERENTIAL EQUATIONS
Peter L. Simon
Department of Applied Analysis and Computational Mathematics,Institute of Mathematics, Eötvös Loránd University Budapest, and
Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences,Hungary
1 / 24
SIS EPIDEMIC ON A NETWORK
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
2 / 24
SIS EPIDEMIC ON A NETWORK
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
Transitions:
S → I, rate: kτ , k is the number of I neighbours.
I → S, rate: γ
2 / 24
SIS EPIDEMIC ON A NETWORK
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
Transitions:S → I, rate: kτ , k is the number of I neighbours.I → S, rate: γ
I
S
SI
recovery
γinfection
τ
2 / 24
SIS EPIDEMIC ON A NETWORK
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
Transitions:
S → I, rate: kτ , k is the number of I neighbours.
I → S, rate: γ
AIM: Derive a simple system of differential equations yielding theexpected number of infected nodes [I](t).
2 / 24
SIS EPIDEMIC ON A NETWORK
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
Transitions:
S → I, rate: kτ , k is the number of I neighbours.
I → S, rate: γ
AIM: Derive a simple system of differential equations yielding theexpected number of infected nodes [I](t).
Known models:
Master equation
Mean-field equation
Pairwise model
Compact pairwise model
...2 / 24
GENERAL MATHEMATICAL MODEL
A graph with N nodes is given
3 / 24
GENERAL MATHEMATICAL MODEL
A graph with N nodes is given
The nodes can be in the states {a1, a2, . . . am}.
3 / 24
GENERAL MATHEMATICAL MODEL
A graph with N nodes is given
The nodes can be in the states {a1, a2, . . . am}.
The state space of the graph has mN elements
3 / 24
GENERAL MATHEMATICAL MODEL
A graph with N nodes is given
The nodes can be in the states {a1, a2, . . . am}.
The state space of the graph has mN elements
The transitions between different states can be described by aPoisson process
Probability of a transition from state ai to state aj in a time interval oflength ∆t is:
1 − exp(−λij∆t).
3 / 24
NETWORK PROCESSES
SIS epidemic
4 / 24
NETWORK PROCESSES
SIS epidemic
States of the nodes: {S, I}.
4 / 24
NETWORK PROCESSES
SIS epidemic
States of the nodes: {S, I}.
Transitions and their rates
S → I, λ = kτ , k is the number of I neighbours.
I → S, λ = γ
4 / 24
NETWORK PROCESSES
SIR epidemic
5 / 24
NETWORK PROCESSES
SIR epidemic
States of the nodes: {S, I,R}.
5 / 24
NETWORK PROCESSES
SIR epidemic
States of the nodes: {S, I,R}.
Transitions and their rates
S → I, λ = kτ , k is the number of I neighbours.
I → R, λ = γ
5 / 24
NETWORK PROCESSES
Rumour spreading
6 / 24
NETWORK PROCESSES
Rumour spreading
States of the nodes: {X ,Y ,Z} (ignorant, spreader, stifler).
6 / 24
NETWORK PROCESSES
Rumour spreading
States of the nodes: {X ,Y ,Z} (ignorant, spreader, stifler).
Transitions and their rates
X → Y , λ = kτ , k is the number of Y neighbours.
Y → Z , λ = γ + jp, j is the number of Y and Z neighbours.
6 / 24
NETWORK PROCESSES
Propagation of activity in neuronal networks
7 / 24
NETWORK PROCESSES
Propagation of activity in neuronal networks
States of the nodes: {E+,E−, I+, I−} (active and inactive excitatory
neurons, active and inactive inhibitory neurons).
7 / 24
NETWORK PROCESSES
Propagation of activity in neuronal networks
States of the nodes: {E+,E−, I+, I−} (active and inactive excitatory
neurons, active and inactive inhibitory neurons).
Transitions and their rates
E+ → E−
, λ = α.
E−→ E+, λ = tanh(iwE − jwI + hE), i, j is the number of E+ and
I+ neighbours.
I+ → I−
, λ = α.
I−→ I+, λ = tanh(iwE − jwI + hI), i, j is the number of E+ and I+
neighbours.
7 / 24
AIM OF THE RESEARCH
Derive differential equations for different processes and for differenttypes of graphs.
8 / 24
AIM OF THE RESEARCH
Derive differential equations for different processes and for differenttypes of graphs.
Frequently used random graphs:
Erdos-Rényi
Configuration model (Bollobás)
Small-world (Watts-Strogatz)
Graphs with scale free degree distribution (Barabási-Albert)
8 / 24
AIM OF THE RESEARCH
Derive differential equations for different processes and for differenttypes of graphs.
Frequently used random graphs:
Erdos-Rényi
Configuration model (Bollobás)
Small-world (Watts-Strogatz)
Graphs with scale free degree distribution (Barabási-Albert)
Examples for network processes:
Epidemic propagation
Rumour spreading
Propagation of neuronal activity
8 / 24
MARKOV CHAIN FOR SIS EPIDEMIC
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
9 / 24
MARKOV CHAIN FOR SIS EPIDEMIC
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
Transitions:
S → I, rate: kτ , k is the number of I neighbours.
I → S, rate: γ
9 / 24
MARKOV CHAIN FOR SIS EPIDEMIC
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
State space for a triangle
I I
I
S I
I
I S
I
I I
S
S S
I
S I
S
I S
S
S S
S
9 / 24
MARKOV CHAIN FOR SIS EPIDEMIC
A graph with N nodes is given
The nodes can be susceptible (S) or infected (I)
State space for a triangle
I I
I
S I
I
I S
I
I I
S
S S
I
S I
S
I S
S
S S
S
Infection: SIS → SII, IIS
Recovery: SIS → SSS
9 / 24
SIS EPIDEMIC
Master equations
XSSS = γ(XSSI + XSIS + XISS),
XSSI = γ(XSII + XISI)− (2τ + γ)XSSI ,
XSIS = γ(XSII + XIIS)− (2τ + γ)XSIS ,
XISS = γ(XISI + XIIS)− (2τ + γ)XISS ,
XSII = γXIII + τ(XSSI + XSIS)− 2(τ + γ)XSII ,
XISI = γXIII + τ(XSSI + XISS)− 2(τ + γ)XISI ,
XIIS = γXIII + τ(XSIS + XISS)− 2(τ + γ)XIIS ,
XIII = −3γXIII + 2τ(XSII + XISI + XIIS),
10 / 24
SIS EPIDEMIC
Master equations
XSSS = γ(XSSI + XSIS + XISS),
XSSI = γ(XSII + XISI)− (2τ + γ)XSSI ,
XSIS = γ(XSII + XIIS)− (2τ + γ)XSIS ,
XISS = γ(XISI + XIIS)− (2τ + γ)XISS ,
XSII = γXIII + τ(XSSI + XSIS)− 2(τ + γ)XSII ,
XISI = γXIII + τ(XSSI + XISS)− 2(τ + γ)XISI ,
XIIS = γXIII + τ(XSIS + XISS)− 2(τ + γ)XIIS ,
XIII = −3γXIII + 2τ(XSII + XISI + XIIS),
2N equations for a graph with N nodes
10 / 24
SIS EPIDEMIC
Master equations
XSSS = γ(XSSI + XSIS + XISS),
XSSI = γ(XSII + XISI)− (2τ + γ)XSSI ,
XSIS = γ(XSII + XIIS)− (2τ + γ)XSIS ,
XISS = γ(XISI + XIIS)− (2τ + γ)XISS ,
XSII = γXIII + τ(XSSI + XSIS)− 2(τ + γ)XSII ,
XISI = γXIII + τ(XSSI + XISS)− 2(τ + γ)XISI ,
XIIS = γXIII + τ(XSIS + XISS)− 2(τ + γ)XIIS ,
XIII = −3γXIII + 2τ(XSII + XISI + XIIS),
The size of the system can be reduced by using the automorphismsof the graph:
Simon, P.L., Taylor, M., Kiss., I.Z., Exact epidemic models on graphs using graph-automorphism
driven lumping, J. Math. Biol., 62 (2011).
10 / 24
MEAN-FIELD APPROXIMATION FOR SIS EPIDEMIC
Exact equation: ˙[I] = τ [SI] − γ[I]
11 / 24
MEAN-FIELD APPROXIMATION FOR SIS EPIDEMIC
Exact equation: ˙[I] = τ [SI] − γ[I]
[SI](t): expected number of SI edges
This differential equation holds for any graphSimon, P.L., Taylor, M., Kiss., I.Z., Exact epidemic models on graphs using graph-automorphism
driven lumping, J. Math. Biol. 62 (2011), 479-508.
11 / 24
MEAN-FIELD APPROXIMATION FOR SIS EPIDEMIC
Exact equation: ˙[I] = τ [SI] − γ[I]
Approximation [SI] ≈ n [I]N [S], where the average degree is n
11 / 24
MEAN-FIELD APPROXIMATION FOR SIS EPIDEMIC
Exact equation: ˙[I] = τ [SI] − γ[I]
Approximation [SI] ≈ n [I]N [S], where the average degree is n
Approximating differential equation for [I]
˙I = τnN
I(N − I)− γ I.
11 / 24
MEAN-FIELD APPROXIMATION FOR SIS EPIDEMIC
Exact equation: ˙[I] = τ [SI] − γ[I]
Approximation [SI] ≈ n [I]N [S], where the average degree is n
Approximating differential equation for [I]
˙I = τnN
I(N − I)− γ I.
This is the well-known compartmental model, which does not giveaccurate result for networks.Reason: the approximation assumes random distribution of infectednodes.
11 / 24
MEAN-FIELD APPROXIMATION FOR SIS EPIDEMIC
Exact equation: ˙[I] = τ [SI] − γ[I]
Approximation [SI] ≈ n [I]N [S], where the average degree is n
Approximating differential equation for [I]
˙I = τnN
I(N − I)− γ I.
This is the well-known compartmental model, which does not giveaccurate result for networks.Reason: the approximation assumes random distribution of infectednodes.
Better idea: derive a differential equation for [SI], this leaded to thepairwise model.Keeling, M.J., The effects of local spatial structure on epidemiological invasions, Proc. R. Soc.
Lond. B 266 (1999), 859-867.
11 / 24
PAIRWISE APPROXIMATION
Keep the exact equation ˙[I] = τ [SI] − γ[I]
and derive a differential equation for [SI].
12 / 24
PAIRWISE APPROXIMATION
Keep the exact equation ˙[I] = τ [SI] − γ[I]
and derive a differential equation for [SI].
Exact differential equations:
˙[I] = τ [SI]− γ[I],˙[SI] = γ([II]− [SI]) + τ([SSI] − [ISI]− [SI]),˙[II] = −2γ[II] + 2τ([ISI] + [SI]),˙[SS] = 2γ[SI]− 2τ [SSI].
12 / 24
PAIRWISE APPROXIMATION
Keep the exact equation ˙[I] = τ [SI] − γ[I]
and derive a differential equation for [SI].
Exact differential equations:
˙[I] = τ [SI]− γ[I],˙[SI] = γ([II]− [SI]) + τ([SSI] − [ISI]− [SI]),˙[II] = −2γ[II] + 2τ([ISI] + [SI]),˙[SS] = 2γ[SI]− 2τ [SSI].
Approximation:
[ABC] ≈n − 1
n[AB][BC]
[B], n average degree
12 / 24
PAIRWISE APPROXIMATION
Keep the exact equation ˙[I] = τ [SI] − γ[I]
and derive a differential equation for [SI].
Exact differential equations:
˙[I] = τ [SI]− γ[I],˙[SI] = γ([II]− [SI]) + τ([SSI] − [ISI]− [SI]),˙[II] = −2γ[II] + 2τ([ISI] + [SI]),˙[SS] = 2γ[SI]− 2τ [SSI].
Approximation:
[ABC] ≈n − 1
n[AB][BC]
[B], n average degree
M. Taylor, P. L. Simon, D. M. Green, T. House, I. Z. Kiss, From Markovian to pairwise epidemic
models and the performance of moment closure approximations, J. Math. Biol. 64 (2012),
1021-1042.12 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Regular random graph with N = 1000 nodes, average degree n = 20,γ = 1, critical value of τ from compartmental model: τcr = γ/n
13 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Regular random graph with N = 1000 nodes, average degree n = 20,γ = 1, critical value of τ from compartmental model: τcr = γ/n
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
τ = 0.9τcr
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
τ = τcr
0 10 20 30 400.03
0.04
0.05
0.06
0.07
0.08
τ = 1.1τcr
prev
alen
ce
t0 10 20 30 40
0.05
0.1
0.15
0.2
0.25
0.3
τ = 1.5τcr
13 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Regular random graph with N = 1000 nodes, average degree n = 20,γ = 1, critical value of τ from compartmental model: τcr = γ/n
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
τ = 0.9τcr
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
τ = τcr
0 10 20 30 400.03
0.04
0.05
0.06
0.07
0.08
τ = 1.1τcr
prev
alen
ce
t0 10 20 30 40
0.05
0.1
0.15
0.2
0.25
0.3
τ = 1.5τcr
Mean-field: dashed, Pairwise: continuousSimulation (average of 200 runs): grey thick curve
13 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Regular random graph with N = 1000 nodes, average degree n = 20,γ = 1, critical value of τ from compartmental model: τcr = γ/n
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
τ = 0.9τcr
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
τ = τcr
0 10 20 30 400.03
0.04
0.05
0.06
0.07
0.08
τ = 1.1τcr
prev
alen
ce
t0 10 20 30 40
0.05
0.1
0.15
0.2
0.25
0.3
τ = 1.5τcr
τ = τcr ⇔ basic reproduction number R0 = 1.
13 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Bimodal random graph with N = 1000 nodes, average degreen = 20, γ = 1, τ = 2τcr = 2γ/nN/2 nodes have degree d1, N/2 nodes have degree d2.
14 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Bimodal random graph with N = 1000 nodes, average degreen = 20, γ = 1, τ = 2τcr = 2γ/nN/2 nodes have degree d1, N/2 nodes have degree d2.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5d
1=18, d
2=22
t
prev
alen
ce
0 5 100
0.1
0.2
0.3
0.4
0.5
d1=5, d
2=35
14 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Bimodal random graph with N = 1000 nodes, average degreen = 20, γ = 1, τ = 2τcr = 2γ/nN/2 nodes have degree d1, N/2 nodes have degree d2.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5d
1=18, d
2=22
t
prev
alen
ce
0 5 100
0.1
0.2
0.3
0.4
0.5
d1=5, d
2=35
Mean-field: dashed, Pairwise: continuousSimulation (average of 200 runs): grey thick curve
14 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Bimodal random graph with N = 1000 nodes, average degreen = 20, γ = 1, τ = 2τcr = 2γ/nN/2 nodes have degree d1, N/2 nodes have degree d2.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5d
1=18, d
2=22
t
prev
alen
ce
0 5 100
0.1
0.2
0.3
0.4
0.5
d1=5, d
2=35
Reason of inaccuracy: in the closure [ABC] ≈ n−1n
[AB][BC][B] it is
assumed that each node has the same degree n.
14 / 24
COMPACT PAIRWISE APPROXIMATION
There are Nk nodes with degree dk for k = 1, 2, . . . ,K .
15 / 24
COMPACT PAIRWISE APPROXIMATION
There are Nk nodes with degree dk for k = 1, 2, . . . ,K .
[ASI] =K∑
k=1
[ASk I], [ASk I] ≈dk − 1
dk
[ASk ][Sk I][Sk ]
15 / 24
COMPACT PAIRWISE APPROXIMATION
There are Nk nodes with degree dk for k = 1, 2, . . . ,K .
[ASI] =K∑
k=1
[ASk I], [ASk I] ≈dk − 1
dk
[ASk ][Sk I][Sk ]
[Sk ]: expected number of susceptible nodes of degree dk ,[Sk I]: expected number of edges connecting an infected node to asusceptible node of degree dk
15 / 24
COMPACT PAIRWISE APPROXIMATION
There are Nk nodes with degree dk for k = 1, 2, . . . ,K .
[ASI] =K∑
k=1
[ASk I], [ASk I] ≈dk − 1
dk
[ASk ][Sk I][Sk ]
Differential equations are needed for the new unknowns.
15 / 24
COMPACT PAIRWISE APPROXIMATION
There are Nk nodes with degree dk for k = 1, 2, . . . ,K .
[ASI] =K∑
k=1
[ASk I], [ASk I] ≈dk − 1
dk
[ASk ][Sk I][Sk ]
˙[Sk ] = γ[Ik ]− τ [Sk I], k = 1, 2, . . . ,K .
15 / 24
COMPACT PAIRWISE APPROXIMATION
There are Nk nodes with degree dk for k = 1, 2, . . . ,K .
[ASI] =K∑
k=1
[ASk I], [ASk I] ≈dk − 1
dk
[ASk ][Sk I][Sk ]
˙[Sk ] = γ[Ik ]− τ [Sk I], k = 1, 2, . . . ,K .
[Sk A] ≈ [SA]dk [Sk ]
∑Kl=1 dl [Sl ]
15 / 24
COMPACT PAIRWISE APPROXIMATION
There are Nk nodes with degree dk for k = 1, 2, . . . ,K .
[ASI] =K∑
k=1
[ASk I], [ASk I] ≈dk − 1
dk
[ASk ][Sk I][Sk ]
˙[Sk ] = γ[Ik ]− τ [Sk I], k = 1, 2, . . . ,K .
[Sk A] ≈ [SA]dk [Sk ]
∑Kl=1 dl [Sl ]
[ASk I] ≈[AS][SI]dk (dk − 1)[Sk ]
S21
⇒ [ASI] ≈ [AS][SI]S2 − S1
S21
S1 =N∑
k=1dk [Sk ], S2 =
K∑
k=1d2
k [Sk ].
15 / 24
COMPACT PAIRWISE MODEL
˙[Sk ]c = γ[Ik ]c − τdk [Sk ]c[SI]cSs
,
˙[SI]c = γ([II]c − [SI]c) + τ([SS]c − [SI]c)[SI]cP − τ [SI]c ,˙[SS]c = 2γ[SI]c − 2τ [SS]c [SI]cP,
˙[II]c = 2τ [SI]c − 2γ[II]c + 2τ [SI]2cP,
16 / 24
COMPACT PAIRWISE MODEL
˙[Sk ]c = γ[Ik ]c − τdk [Sk ]c[SI]cSs
,
˙[SI]c = γ([II]c − [SI]c) + τ([SS]c − [SI]c)[SI]cP − τ [SI]c ,˙[SS]c = 2γ[SI]c − 2τ [SS]c [SI]cP,
˙[II]c = 2τ [SI]c − 2γ[II]c + 2τ [SI]2cP,
with Ss =∑K
k=1 dk [Sk ]c and P = 1S2
s
K∑
k=1(dk − 1)dk [Sk ]c .
16 / 24
COMPACT PAIRWISE MODEL
˙[Sk ]c = γ[Ik ]c − τdk [Sk ]c[SI]cSs
,
˙[SI]c = γ([II]c − [SI]c) + τ([SS]c − [SI]c)[SI]cP − τ [SI]c ,˙[SS]c = 2γ[SI]c − 2τ [SS]c [SI]cP,
˙[II]c = 2τ [SI]c − 2γ[II]c + 2τ [SI]2cP,
Compact pairwise model: K + 3 equations
16 / 24
COMPACT PAIRWISE MODEL
˙[Sk ]c = γ[Ik ]c − τdk [Sk ]c[SI]cSs
,
˙[SI]c = γ([II]c − [SI]c) + τ([SS]c − [SI]c)[SI]cP − τ [SI]c ,˙[SS]c = 2γ[SI]c − 2τ [SS]c [SI]cP,
˙[II]c = 2τ [SI]c − 2γ[II]c + 2τ [SI]2cP,
Compact pairwise model: K + 3 equations
More complex and accurate models:
Pre-compact pairwise model: 5K equations
16 / 24
COMPACT PAIRWISE MODEL
˙[Sk ]c = γ[Ik ]c − τdk [Sk ]c[SI]cSs
,
˙[SI]c = γ([II]c − [SI]c) + τ([SS]c − [SI]c)[SI]cP − τ [SI]c ,˙[SS]c = 2γ[SI]c − 2τ [SS]c [SI]cP,
˙[II]c = 2τ [SI]c − 2γ[II]c + 2τ [SI]2cP,
Compact pairwise model: K + 3 equations
More complex and accurate models:
Pre-compact pairwise model: 5K equations
Heterogeneous pairwise model: 2K 2 + K equations
16 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Bimodal random graph with N = 1000 nodes, average degreen1 = 20, γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
N/2 nodes have degree d1 = 5, N/2 nodes have degree d2 = 35.
17 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Bimodal random graph with N = 1000 nodes, average degreen1 = 20, γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
N/2 nodes have degree d1 = 5, N/2 nodes have degree d2 = 35.
0 5 10 150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
t
I/N
17 / 24
COMPARISON OF ODE MODELS TO SIMULATION
Bimodal random graph with N = 1000 nodes, average degreen1 = 20, γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
N/2 nodes have degree d1 = 5, N/2 nodes have degree d2 = 35.
0 5 10 150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
t
I/N
Pairwise: dashed, Compact pairwise: continuous black,Heterogeneous pairwise: continuous red,Simulation (average of 200 runs): grey thick curve
17 / 24
SUPER COMPACT PAIRWISE MODEL
Compact pairwise model is accurate for heterogeneous networks, butit contains K + 3 differential equations.
18 / 24
SUPER COMPACT PAIRWISE MODEL
Compact pairwise model is accurate for heterogeneous networks, butit contains K + 3 differential equations.
For power-law random graphs or Barabási-Albert networks K is large.
18 / 24
SUPER COMPACT PAIRWISE MODEL
Compact pairwise model is accurate for heterogeneous networks, butit contains K + 3 differential equations.
AIM: derive a system of 4 differential equations performing well forstrongly heterogeneous networks.
18 / 24
SUPER COMPACT PAIRWISE MODEL
Compact pairwise model is accurate for heterogeneous networks, butit contains K + 3 differential equations.
AIM: derive a system of 4 differential equations performing well forstrongly heterogeneous networks.
˙[Sk ]c = γ[Ik ]c − τdk [Sk ]c[SI]cSs
,
˙[SI]c = γ([II]c − [SI]c) + τ([SS]c − [SI]c)[SI]cP − τ [SI]c ,˙[SS]c = 2γ[SI]c − 2τ [SS]c [SI]cP,
˙[II]c = 2τ [SI]c − 2γ[II]c + 2τ [SI]2cP,
with Ss =∑K
k=1 dk [Sk ]c and P = 1S2
s
K∑
k=1(dk − 1)dk [Sk ]c .
18 / 24
SUPER COMPACT PAIRWISE MODEL
Compact pairwise model is accurate for heterogeneous networks, butit contains K + 3 differential equations.
AIM: derive a system of 4 differential equations performing well forstrongly heterogeneous networks.
IDEA: Approximate S2 =K∑
k=1d2
k [Sk ] without having differential
equations for [Sk ] and use the simple pairwise model with the closure[ASI] ≈ [AS][SI]S2−S1
S21
.
18 / 24
SUPER COMPACT PAIRWISE MODEL
Compact pairwise model is accurate for heterogeneous networks, butit contains K + 3 differential equations.
AIM: derive a system of 4 differential equations performing well forstrongly heterogeneous networks.
IDEA: Approximate S2 =K∑
k=1d2
k [Sk ] without having differential
equations for [Sk ] and use the simple pairwise model with the closure[ASI] ≈ [AS][SI]S2−S1
S21
.
˙[S] = γ[I]− τ [SI],˙[SI] = γ([II]− [SI]) + τ([SSI] − [ISI]− [SI]),˙[II] = −2γ[II] + 2τ([ISI] + [SI]),˙[SS] = 2γ[SI]− 2τ [SSI].
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APPROXIMATION OF THE SECOND MOMENT S2
[S] =K∑
k=1[Sk ],
K∑
k=1dk [Sk ] = [SI] + [SS]
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APPROXIMATION OF THE SECOND MOMENT S2
[S] =K∑
k=1[Sk ],
K∑
k=1dk [Sk ] = [SI] + [SS]
Let sk = [Sk ]/[S], then
s1 + s2 + . . .+ sK = 1,
d1s1 + d2s2 + . . .+ dK sK = nS :=[SI] + [SS]
[S].
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APPROXIMATION OF THE SECOND MOMENT S2
[S] =K∑
k=1[Sk ],
K∑
k=1dk [Sk ] = [SI] + [SS]
Let sk = [Sk ]/[S], then
s1 + s2 + . . .+ sK = 1,
d1s1 + d2s2 + . . .+ dK sK = nS :=[SI] + [SS]
[S].
Degree distribution of the network: pk = Nk/N.
19 / 24
APPROXIMATION OF THE SECOND MOMENT S2
[S] =K∑
k=1[Sk ],
K∑
k=1dk [Sk ] = [SI] + [SS]
Let sk = [Sk ]/[S], then
s1 + s2 + . . .+ sK = 1,
d1s1 + d2s2 + . . .+ dK sK = nS :=[SI] + [SS]
[S].
Degree distribution of the network: pk = Nk/N.
Numerical observation: sk/pk is a linear function of the degree dk .
19 / 24
APPROXIMATION OF THE SECOND MOMENT S2
[S] =K∑
k=1[Sk ],
K∑
k=1dk [Sk ] = [SI] + [SS]
Let sk = [Sk ]/[S], then
s1 + s2 + . . .+ sK = 1,
d1s1 + d2s2 + . . .+ dK sK = nS :=[SI] + [SS]
[S].
Degree distribution of the network: pk = Nk/N.
Numerical observation: sk/pk is a linear function of the degree dk .
Then some calculation yields
(n2 − n21)
sk
pk= n2 − nSn1 + dk (nS − n1),
where ni =∑
d ik pk . Leading to
19 / 24
APPROXIMATION OF THE SECOND MOMENT S2
[S] =K∑
k=1[Sk ],
K∑
k=1dk [Sk ] = [SI] + [SS]
Let sk = [Sk ]/[S], then
s1 + s2 + . . .+ sK = 1,
d1s1 + d2s2 + . . .+ dK sK = nS :=[SI] + [SS]
[S].
Degree distribution of the network: pk = Nk/N.
Numerical observation: sk/pk is a linear function of the degree dk .
(n2 − n21)
K∑
k=1
d2k sk = n2(n2 − nSn1) + n3(nS − n1),
where ni =∑
d ik pk .
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APPROXIMATION OF THE SECOND MOMENT S2
S2 =
K∑
k=1
d2k [Sk ] ≈
K∑
k=1
d2k [S]sk = [S]
n2(n2 − nSn1) + n3(nS − n1)
n2 − n21
.
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APPROXIMATION OF THE SECOND MOMENT S2
S2 =
K∑
k=1
d2k [Sk ] ≈
K∑
k=1
d2k [S]sk = [S]
n2(n2 − nSn1) + n3(nS − n1)
n2 − n21
.
S1 = [SI] + [SS] = nS [S] implies
S2 − S1
S21
≈1
n2S [S]
(
n2(n2 − nSn1) + n3(nS − n1)
n2 − n21
− nS
)
.
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APPROXIMATION OF THE SECOND MOMENT S2
S2 =
K∑
k=1
d2k [Sk ] ≈
K∑
k=1
d2k [S]sk = [S]
n2(n2 − nSn1) + n3(nS − n1)
n2 − n21
.
S1 = [SI] + [SS] = nS [S] implies
S2 − S1
S21
≈1
n2S [S]
(
n2(n2 − nSn1) + n3(nS − n1)
n2 − n21
− nS
)
.
The new closure relation:
[ASI] ≈ [AS][SI]S2 − S1
S21
=[AS][SI]
nS [S]
(
n2(n2 − nSn1) + n3(nS − n1)
nS(n2 − n21)
− 1)
.
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SUPER COMPACT PAIRWISE MODEL
˙[S]s = γ[I]s − τ [SI]s ,˙[SI]s = γ([II]s − [SI]s) + τ [SI]s([SS]s − [SI]s)Q − τ [SI]s ,˙[SS]s = 2γ[SI]s − 2τ [SI]s[SS]sQ,
˙[II]s = −2γ[II]s + 2τ [SI]2sQ + 2τ [SI]s ,
where
Q =1
nS [S]
(
n2(n2 − nSn1) + n3(nS − n1)
nS(n2 − n21)
− 1)
, nS :=[SI] + [SS]
[S],
ni =∑
d ik pk .
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PERFORMANCE OF THE SUPER COMPACT PAIRWISE
MODEL
Bimodal random graph with N = 1000 nodes, average degreen1 = 20, γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
N1 nodes have degree d1 = 5, N2 nodes have degree d2 = 35.
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PERFORMANCE OF THE SUPER COMPACT PAIRWISE
MODEL
Bimodal random graph with N = 1000 nodes, average degreen1 = 20, γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
N1 nodes have degree d1 = 5, N2 nodes have degree d2 = 35.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t
I/N
22 / 24
PERFORMANCE OF THE SUPER COMPACT PAIRWISE
MODEL
Bimodal random graph with N = 1000 nodes, average degreen1 = 20, γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
N1 nodes have degree d1 = 5, N2 nodes have degree d2 = 35.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t
I/N
PW: dashed, compact PW: continuous, super compact PW: circles,upper curves: N1 = 0.1N, N2 = 0.9N, middle curves: N1 = 0.5N,N2 = 0.5N, lower curves: N1 = 0.9N, N2 = 0.1N
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PERFORMANCE OF THE SUPER COMPACT PAIRWISE
MODEL
Power-law random graph with N = 1000 nodes, pk = Ck−α fork = kmin, kmin + 1, . . . , kmax , γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
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PERFORMANCE OF THE SUPER COMPACT PAIRWISE
MODEL
Power-law random graph with N = 1000 nodes, pk = Ck−α fork = kmin, kmin + 1, . . . , kmax , γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
t
I/N
23 / 24
PERFORMANCE OF THE SUPER COMPACT PAIRWISE
MODEL
Power-law random graph with N = 1000 nodes, pk = Ck−α fork = kmin, kmin + 1, . . . , kmax , γ = 1, τ = 3γn1/n2, ni =
∑
d ik pk
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
t
I/N
PW: dashed, compact PW: continuous, super compact PW: circles,upper curves: kmin = 10, kmax = 140,lower curves: kmin = 5, kmax = 30
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Thank you for your attention!
Simon, P.L., Kiss., I.Z., Super compact pairwise model for SIS epidemic on heterogeneous
networks, J. Complex Networks (2015).
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